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Conjugate Hyperbola

Let A 0 A 1...

Question

Let $A_{0} A_{1} A_{2} A_{3} A_{4} A_{5}$ be a regular hexagon inscribed in a circle unit radius. Then the product of the lengths of the line segments $A_{0} A_{1}, A_{0} A_{2}$ and $A_{0} A_{4}$ is

A

\frac{3}{4}

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B

3 \sqrt{3}

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C

3

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D

\frac{3 \sqrt{3}}{2}

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Solution

The correct option is C $3$
We have ${(A_{0} A_{1})}^{2} = \frac{1}{4} + \frac{3}{4} = 1 \Rightarrow A_{0} A_{1} = 1$
$A_{0} A_{2} = {(\frac{3}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2} = \frac{9 + 3}{4} = 3 \Rightarrow A_{0} A_{2} = \sqrt{3}$
Similarly $A_{0} A_{4} = \sqrt{3}$
Thus $(A_{0} A_{1}) (A_{0} A_{2}) (A_{0} A_{4}) = 3$

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